Exactly solvable quantum few-body systems associated with the symmetries of the three-dimensional and four-dimensional icosahedra

TL;DR

This paper constructs new exactly solvable quantum few-body systems based on the symmetries of 3D and 4D icosahedral groups, expanding the class of solvable models in quantum mechanics.

Contribution

It introduces a family of solvable four-body quantum systems linked to non-crystallographic reflection groups, specifically the icosahedron and 600-cell symmetries.

Findings

Explicit construction of a one-parametric family of four-body systems

Demonstration of systems constrained to a half-line in certain limits

Extension of solvable models to 4D icosahedral symmetry

Abstract

The purpose of this article is to demonstrate that non-crystallographic reflection groups can be used to build new solvable quantum particle systems. We explicitly construct a one-parametric family of solvable four-body systems on a line, related to the symmetry of a regular icosahedron: in two distinct limiting cases the system is constrained to a half-line. We repeat the program for a 600-cell, a four-dimensional generalization of the regular three-dimensional icosahedron.